Fraction

“ Continued Fractions.—The theory of continued fractions (q.v.) gives a method of expressing a number, in certain cases, as a continued product. A continued fraction, of the kind we are considering, is an expression of the form where b, c, d, . . . are integers, and a is an integer or zero. ”

“ Every number or fraction is a multiple-part of every other number or fraction. If, for example, we ask what part 12 is of 7, we see that on dividing 7 into 7 parts, and repeating one of these parts 12 times, we obtain 12 ”

“ The Evaluation of Continued Fractions.—The numerators and denominators of the convergents to the general continued fraction both satisfy the difference equation un = anun-1 + bnun-2. When we can solve this equation we have an expression for the nth convergent to the fraction, generally in the form of the quotient of two series, each of n terms. ”

“ The fifth view is nothing more than a change of words: A number reduced to 7/9 of its amount has every 9 converted into a 7, and any fraction of a 9 which may remain over into the corresponding fraction of 7. This is completely proved when we prove the equation 7/9 of a = 7 times a/9. The sixth, seventh, and eighth views are illustrated in the chapter on proportion.
When the student comes to algebra, he will find that, in all the applications of that science, fractions such as a/b most frequently require that a and b should be themselves supposed to be fractions. ”

“ If one reduces the fraction whose numerator is unity and whose denominator is unity plus a variable, in a series prescribed by the ratio to the powers of this variable, it is easy to see that in supposing the variable equal to unity the fraction becomes 1⁄2, and the series becomes plus one, minus one, plus one, minus one, etc. ”

“ Then, if there be any figures before the point, they make the whole number which the fraction contains. The first figure after the point with the denominator 10, the second with the denominator 100, and so on, are the fractions of which the first fraction is composed.
135. Decimal fractions are not usually written at full length. It is more convenient to write the numerator only, and to cut off from the numerator as many figures as there are ciphers in the denominator, when that is possible, by a point. ”

“ Thus, if the ratio of the less to the greater period is expressed by the fraction 1⁄2, where the difference between the numerator and the denominator is one, the commensurability is of the first order ”

“ If fraction does not exist in the Natural Order, still less shall we find it in the Moral Order, where ideas and sentiments may be as varied as the species of the Vegetable kingdom and yet be always whole. The theory of fractions is therefore another signal instance of the servility of your mind.
“Thus Number, with its infinite minuteness and its infinite expansion, is a power whose weakest side is known to you, but whose real import escapes your perception. ”

“ The true definition of a fraction may be thus summed up: It is the division of its numerator by its denominator; as seven-eighths are equivalent to the whole number 7 divided by 8—whence a fraction is obtained. Decimal fractions simplify calculations greatly, as they are constructed on the principle of having one common denominator—a multiple of 10; and thus fractions can be added, subtracted, and divided without repeating over the tedious process of bringing down to a common denominator. See Arithmetic; Decimal Fraction. ”

“ A class is engaged in deriving inductively the generalization that multiplying the numerator of a fraction by any number multiplies the fraction by that number; will there be any occasion for deductive thinking as the work proceeds? ”

“ The Conversion of Series and Products into Continued Fractions. 1. A continued fraction may always be found whose nth convergent shall be equal to the sum to n terms of a given series or the product to n factors of a given continued product. In fact, a continued fraction
can be constructed having for the numerators of its successive convergents any assigned quantities p1, p2, p3, . . ., pn, and for their denominators any assigned quantities q1, q2, q3, . . ”

“ The value of this new fraction for the assigned value of the variable will be the limiting value of the original fraction.
A rigorous proof of this rule is beyond the scope of this book and is left for more advanced treatises.
Illustrative Example 1. Evaluate for x = 0.
Solution. ∴ indeterminate. ∴ indeterminate. Ans. 113. Evaluation of the indeterminate form . If a function takes on the indeterminate form for , we write the given function
so as to cause it to take on one of the forms or , thus bringing it under §111 or §112. ”

“ Though not elementary algebra, it is within the reach of a student of ordinary books. [674]
Let a continued fraction, such as
a —— b + c —— d + e - f + etc., be abbreviated into a/b+ c/d+ e/f+ etc.: each fraction being understood as falling down to the side of the preceding sign +. In every such fraction we may suppose b, d, f, etc. positive; a, c, e, &c. being as required: and all are supposed integers. If this succession be continued ad infinitum, and if a/b, c/d, e/f, etc. all lie between -1 and +1, exclusive, the limit of the fraction must be incommensurable with unity ”

“ Every convergent is nearer to the value of the whole fraction than any preceding convergent.
Every convergent is a nearer approximation to the value of the whole fraction than any fraction whose denominator is less than that of the convergent. ”

“ Devices and illustrations of a material kind are necessary to build up in the pupil’s mind at the beginning a clear concept of a tenth, etc., etc., and then to show that one-tenth written as a decimal is only a shorthand way of writing 1/10 as a common fraction, and so on. He sees very soon that the decimal is only a shorthand common fraction, and this notion he must hold to. This is the vital point in decimals. The idea that they can be changed into common fractions and the reverse at will establishes the fact in the pupil’s mind that they are common fractions and not uncommon ones. ”

“ In teaching arithmetic there does not exist any greater difficulty in getting small children to grasp the nature of the fraction as such than in getting them to grasp the idea of the simpler whole numbers. It is true that the fractions 1⁄2, 1⁄3, 1⁄4, etc., as symbols, are a little more complex than are the single digits; but as to the real meaning, when once the fractional idea has been properly developed by the teacher and the significance of the idea apprehended by the pupil, it is as easily understood as any other simple truth. ”

“ If we divide 9 into 9 equal parts, each is 1, and 7 of the parts are 7; consequently the fraction which 7 is of 9 is 7/9. The fourth view follows immediately: For a time is only a word used to express one of the repetitions which take place in multiplication, and we allow ourselves, by an easy extension of language, to speak of a portion of a number as being that number taken a part of a time. ”

“ As there is 2⁄5 in the denominator, we must take the part of which 2⁄5 will be a unit. That part is larger than a unit; it is 21⁄2 units; 21⁄2 is that of which 2⁄5 is 1. The above fraction then directs us to repeat 21⁄2 units 31⁄4 times. By extending our word ‘multiplication’ to the taking of a part of a time, all multiplications are also divisions, and all divisions multiplications, and all the terms connected with either are subject to be applied to the results of the other.
If 21⁄3 yards cost 31⁄2 shillings, how much does one yard cost? ”

“ After thousandths, come ten thousandths, hundred thousandths, mil- Konths, &c. but for ordinary uses we seldom come down to so small a fraction. Multiplication is performed as if there were no comma ; but in the sum total, as many figures must be cut off at the right hand of the comma, as are cut off in both the multiplier and sum multiplied. ”

“ Not only is there a mathematical progression, as the lost fraction increases from one-hundredth to one-tenth, and then to one-half of the whole, but the intrinsic value of the loss rises in a corresponding series. ”

“ Lemma 2.—The constant term in any proper algebraical fraction developed in ascending powers of its variable is the same as the residue, with changed sign, of the sum of the fractions obtained by substituting in the given fraction, in lieu of the variable, its exponential multiplied in succession by each of its values (zero excepted, if there be such) , which makes the given fraction infinite. ”

“ Case III. When, among the factors of the denominator there are some which are raised to some power, one must allow for the possible existence of partial fractions having for denominator the several powers of that factor up to the highest. For instance, in splitting the fraction we must allow for the possible existence of a denominator as well as and . ”

“ It appears at first paradoxical, but it is well known to all mathematicians, that, through all eternity, fractions may be taken from a weight without the weight ever being annihilated. It is sufficient that each successive fraction be less than the preceding one, in a determined and regular proportion. ”

“ The modern method is to deal with fractions which have 100 as denominator; such fractions are called percentages. They only apply accurately to divisions by 2, 4, 5, 10, 20, 25 or 50; but they have the convenience of fitting in with the denary scale of notation, and they can be extended to other divisions by using a mixed number as numerator. ”

“ In Arithmetic, the various operations in addition, subtraction, multiplication, and division, reduction, simple and compound proportion, and vulgar and decimal fractions, must be thoroughly understood and readily performed. ”

“ Now this process can certainly be reversed: that is to say, that if this last expression were given, it is certain that it can somehow be split back again into its original components or partial fractions. Only we do not know in every case that may be presented to us how we can so split it. In order to find this out we shall consider a simple case at first. But it is important to bear in mind that all which follows applies only to what are called “proper” algebraic fractions, meaning fractions like the above, which have the numerator of a lesser degree than the denominator ”

“ In the case of decimal fractions the psychological process is more complex still; for the pupil has given him one of the terms, the numerator, from which 20 he must mentally deduce the denominator from the position of the decimal point. This doubles the work of reading and recognizing the fractional number. But it makes addition and subtraction of fractions nearly as easy as that of simple numbers and assists also in multiplication of fractions. But division of decimals is a much more complex operation than that of common fractions. ”

“ Aim. Readiness in the four mathematical operations in whole numbers, decimal numbers, and simple fractions, written and oral. Ability to independently solve various examples in the forms in which they generally appear in practical life, also examples in proportion and examples in percentage, together with problems of planes and solids. Course and mode of its execution as given in a text—Instruction in Mathematics (Regneundervisningen) —followed in all essentials. ”

“ What has been found to be more easily understood precedes the more difficult, without respect to its scientific relation. Fractions present no greater difficulty than wholes, so they accompany the teaching of integral numbers from the beginning. ”

“ A fractional amount is made to refer to the greater, and not the lesser number; as, klockan är tre qvarter på fyra, 'it is a quarter (three-quarters on) to four;' klockan half fem, 'half-past four;' half annan, 'one and a half.' Where och, 'and,' is used, the position of the fractional part is altered in such sentences ”

“ Children get the idea of half, third, or quarter of many things long before they enter school, and they will as readily learn to add, subtract, multiply, and divide fractions as they will whole numbers. In using fractions they will draw diagrams and pictures representing the processes of work as quickly and easily as they illustrate similar work with integers. ”

“ To multiply two fractions together, multiply the numerators for a new numerator and the denominators for a new denominator.
As in Example 2 the work is shortened if we cancel common factors from the numerators and denominators. ”

“ If in the natural world there were no other way of getting things save by such successive addition of their logically involved fractions, no complete units or whole things would ever come into being, for the fractions' sum would always leave a remainder. ”

“ Express your values in the metric or decimal system, and you can add or multiply them with great facility; but the difficulty lies in getting those particular fractional values expressed which we have most occasion to use in everyday life, and which it is the instinctive habit of our minds to deal with—such as thirds, fourths, sixths, eighths. ”

“ Let this be called m; then 1/m is the corresponding power of 2⁄5; and a fraction whose denominator can be made as great as we please, can itself be made as small as we please (112) .
197. We have, then, in the series
1 r rr rrr rrrr &c.
[Pg 115] I. A series of increasing terms, if r be greater than 1. II. Of terms having the same value, if r be equal to 1. III. A series of decreasing terms, if r be less than 1. In the first two cases, the sum
1 + r + rr + rrr + &c.
may evidently be made as great as we please, by sufficiently increasing the number of terms. ”